Algorithmic Superactivation of Asymptotic Quantum Capacity of Zero-Capacity Quantum Channels

The superactivation of zero-capacity quantum channels makes it possible to use two zero-capacity quantum channels with a positive joint capacity for their output. Currently, we have no theoretical background to describe all possible combinations of superactive zero-capacity channels; hence, there may be many other possible combinations. In practice, to discover such superactive zero-capacity channel-pairs, we must analyze an extremely large set of possible quantum states, channel models, and channel probabilities. There is still no extremely efficient algorithmic tool for this purpose. This paper shows an efficient algorithmical method of finding such combinations. Our method can be a very valuable tool for improving the results of fault-tolerant quantum computation and possible communication techniques over very noisy quantum channels.Comment: 35 pages, 17 figures, Journal-ref: Information Sciences (Elsevier, 2012), presented in part at Quantum Information Processing 2012 (QIP2012), v2: minor changes, v3: published version; Information Sciences, Elsevier, ISSN: 0020-0255; 201

Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms

We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques

Privacy Amplification via Importance Sampling

We examine the privacy-enhancing properties of subsampling a data set via importance sampling as a pre-processing step for differentially private mechanisms. This extends the established privacy amplification by subsampling result to importance sampling where each data point is weighted by the reciprocal of its selection probability. The implications for privacy of weighting each point are not obvious. On the one hand, a lower selection probability leads to a stronger privacy amplification. On the other hand, the higher the weight, the stronger the influence of the point on the output of the mechanism in the event that the point does get selected. We provide a general result that quantifies the trade-off between these two effects. We show that heterogeneous sampling probabilities can lead to both stronger privacy and better utility than uniform subsampling while retaining the subsample size. In particular, we formulate and solve the problem of privacy-optimal sampling, that is, finding the importance weights that minimize the expected subset size subject to a given privacy budget. Empirically, we evaluate the privacy, efficiency, and accuracy of importance sampling-based privacy amplification on the example of k-means clustering.Comment: Under review for NeurIPS 202

Data Summarizations for Scalable, Robust and Privacy-Aware Learning in High Dimensions

The advent of large-scale datasets has offered unprecedented amounts of information for building statistically powerful machines, but, at the same time, also introduced a remarkable computational challenge: how can we efficiently process massive data? This thesis presents a suite of data reduction methods that make learning algorithms scale on large datasets, via extracting a succinct model-specific representation that summarizes the full data collection—a coreset. Our frameworks support by design datasets of arbitrary dimensionality, and can be used for general purpose Bayesian inference under real-world constraints, including privacy preservation and robustness to outliers, encompassing diverse uncertainty-aware data analysis tasks, such as density estimation, classification and regression. We motivate the necessity for novel data reduction techniques in the first place by developing a reidentification attack on coarsened representations of private behavioural data. Analysing longitudinal records of human mobility, we detect privacy-revealing structural patterns, that remain preserved in reduced graph representations of individuals’ information with manageable size. These unique patterns enable mounting linkage attacks via structural similarity computations on longitudinal mobility traces, revealing an overlooked, yet existing, privacy threat. We then propose a scalable variational inference scheme for approximating posteriors on large datasets via learnable weighted pseudodata, termed pseudocoresets. We show that the use of pseudodata enables overcoming the constraints on minimum summary size for given approximation quality, that are imposed on all existing Bayesian coreset constructions due to data dimensionality. Moreover, it allows us to develop a scheme for pseudocoresets-based summarization that satisfies the standard framework of differential privacy by construction; in this way, we can release reduced size privacy-preserving representations for sensitive datasets that are amenable to arbitrary post-processing. Subsequently, we consider summarizations for large-scale Bayesian inference in scenarios when observed datapoints depart from the statistical assumptions of our model. Using robust divergences, we develop a method for constructing coresets resilient to model misspecification. Crucially, this method is able to automatically discard outliers from the generated data summaries. Thus we deliver robustified scalable representations for inference, that are suitable for applications involving contaminated and unreliable data sources. We demonstrate the performance of proposed summarization techniques on multiple parametric statistical models, and diverse simulated and real-world datasets, from music genre features to hospital readmission records, considering a wide range of data dimensionalities.Nokia Bell Labs, Lundgren Fund, Darwin College, University of Cambridge Department of Computer Science & Technology, University of Cambridg

Solving kk-means on High-dimensional Big Data

In recent years, there have been major efforts to develop data stream algorithms that process inputs in one pass over the data with little memory requirement. For the $k$-means problem, this has led to the development of several $(1+\varepsilon)$-approximations (under the assumption that $k$ is a constant), but also to the design of algorithms that are extremely fast in practice and compute solutions of high accuracy. However, when not only the length of the stream is high but also the dimensionality of the input points, then current methods reach their limits. We propose two algorithms, piecy and piecy-mr that are based on the recently developed data stream algorithm BICO that can process high dimensional data in one pass and output a solution of high quality. While piecy is suited for high dimensional data with a medium number of points, piecy-mr is meant for high dimensional data that comes in a very long stream. We provide an extensive experimental study to evaluate piecy and piecy-mr that shows the strength of the new algorithms.Comment: 23 pages, 9 figures, published at the 14th International Symposium on Experimental Algorithms - SEA 201